In this section, we focus on the compressible Euler limit from the scaled BFD equation (REF ). On the basis of the \(L^2-L^\infty \)  approach [1]}, [2]}, it suffices to estimate norms \(\Vert f_{R,\varepsilon }\Vert _2\)  and \(\Vert h_{R,\varepsilon }\Vert _\infty \)  for the equation (REF ) of the reminder \(F_{R,\varepsilon }\) . Here, \(f_{R,\varepsilon }\)  and \(h_{R,\varepsilon }\)  are given by (REF ) and (REF ) respectively, i.e.
\(f_{R,\varepsilon }=\frac{F_{R,\varepsilon }}{\sqrt{\mu (1-\mu )}},\quad h_{R,\varepsilon }=w(v)\frac{F_{R,\varepsilon }}{\sqrt{\mu _F(1-\mu _F)}},\quad \mu _F=\frac{1}{1+e^{\frac{|v|^2}{2T_m}-1}},\quad w(v)=(1+|v|)^l,\,l\geqslant 9.\) 
